Reconfigurable magnon interference by on-chip dynamic wavelength conversion

Spin waves (SWs), an ultra-low power magnetic excitation in ferro or antiferromagnetic media, have tremendous potential as transport less data carriers for post-CMOS technology using their wave interference properties. The concept of magnon interference originates from optical interference, resulting in a historical taboo of maintaining an identical wavevector for magnon interference-based devices. This makes the attainment of on-chip design reconfigurability challenging owing to the difficulty in phase tuning via external fields. Breaking the taboo, this study explores a novel technique to systematically control magnon interference using asymmetric wavevectors from two different SW modes (magnetostatic surface SWs and backward volume magnetostatic SWs) in a microstructured yttrium iron garnet crossbar. Using this system, we demonstrate phase reconfigurability in the interference pattern by modulating the thermal landscape, modifying the dispersion of the interfering SW modes. Thus, we manifest that such a tunable interference can be used to implement reconfigurable logic gates operating between the XNOR and XOR modes by using symmetric and asymmetric interference, respectively.


S4. Prediction of temperature change from the FMR shift
The current-dependent resonance frequency shift is shown in Figure S4(a). The open circles denote the experimental data, which were fitted (dotted line) with the temperature-dependent modified Kittel equation shown below.
Using this equation, we can express the temperature dependent FMR shift as ∆ FMR ( − RT ) ≈ ∆ FMR (∆ ) = FMR ( ) − FMR ( = RT ) . From this current dependent resonance shift, we calculated the temperature change as shown in Figure S4(b). To quantify the current-dependent temperature change, we measured the current-dependent resistance ( ( ) ) of the Pt layer, shown in Figure S4(c) and found that it increases quadratically with applied current, according to ( ) = 0 + 2 2 , where 2 = 45 Ω −2 is the fitting parameter and 0 = 9.3 Ω is the total resistance of the Pt layer at room temperature. Considering the thermal conductivity of YIG as 9 W/m/K and using the Wiedemann-Franz relation, Thiery et. al. analytically modeled the current-induced temperature change as Δ = ( ( ) − 0 )/ 0 , where = 254 K is specific to Pt. Figure  S4(b) represents the model-extracted Δ as a function of 2 .The experiment and fitted curves agreed almost perfectly, and the temperature at the CPW1 region increased in steps of 0.1 K, 0.5 K, 1.1 K, 2.0 K, 3.0 K, 4.5 K, and 6.0 K upon the application of JDC = 1.5, 3.0, 4.5, 6.0, 7.5, 9.0, and 10.5 GAm -2 , respectively.

S5. Generation of the thermal landscape by COMSOL
To map the temperature distribution of the crossbar SW device, the AC/DC module and heat transfer module in COMSOL Multiphysics were adopted for the simulation of heat generation and transfer. The simulated device comprised a crossbar YIG layer for SW propagation with a thickness of 90 nm on a GGG substrate (10 mm long, 10 mm wide, and 0.5 mm thick). Distances depicted in Figure S5 represent the seperations between antennas and heater. A Pt layer (50 nm thick) connected to an Au layer (90 nm thick) was deposited on the YIG, inducing Joule heating when an electric current flowed through it. When in operation, the upper side of the device was in contact with the surrounding air, and the bottom was in contact with the aluminum stage. The electric current and layered shell interface in the AC/DC Module were used to simulate the electricallygenerated heat in the Pt layer when different electric currents (0 A, 0.005 A, 0.01 A, …, 0.065 A, 0.07 A) were applied. The produced heat rate per area in the Pt layer is defined as: where is the thickness of the Pt layer, is the power density, is the current density, is the electric field, is the electrical conductivity, and is the voltage.
The heat transfer in the solid interface of the heat transfer module was used to simulate the heat transfer and temperature distribution in the entire device. After heat generation, the device dissipated heat to the surrounding air on its upper side and the aluminum stage on its bottom side, corresponding to heat flux in the convection process. The heat transfer in the device can be expressed as: where is the density, is the heat capacity at constant pressure, ⃗⃗ is the velocity vector of translational motion, is the temperature, ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ is the heat flux by conduction, ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ is the heat flux by convection, and is the power density of the heat source.
By coupling these two interfaces and setting the boundary conditions, the temperature distribution on the entire device can be obtained using the parameters listed in Supplementary Table 1  From Figure 2(a) in the main text, we observed a drastic change in the ON/OFF ratio (isolation ratio) between the constructive and destructive interference in the dB scale upon application of current. The intensity difference attained a maximum of 21 dB at JDC = 0 GAm -2 , reducing gradually thereafter until it reached 4.5 dB at JDC = 6.7 GAm -2 . At this point, the difference increased gradually with JDC and reached 10 dB at JDC = 10.5 GAm -2 . This can be explained in terms of the difference in incoming SW signal intensity from CPW1 and CPW3, reflected in the scattering parameters S21 and S23, respectively. As shown in Figure S4 (a-h), S21 and S23 represents the SW signals excited in CPW1 and CPW3, respectively, and detected in CPW2. According to the wave superposition theory, the intensity of the interfering waves should be equal to obtain the best contrast between constructive and destructive interference. We chose the 1.685 GHz frequency, as indicated by the red dotted line. As the current density increased, the intensity gap between S21 and S23 widened and reached a maximum difference of approximately 7.5 dB at JDC = 6.7 GAm -2 owing to temperature-induced SW spectral shift. A larger gap between S21 and S23 indicates that a large signal is modulated by a small signal, which weakens the interference effect. Consequently, we observed a small difference in interference gain between constructive and destructive interference in the case of JDC = 6.7 GAm -2 . However, with a further increase in the current density, the gap between S21 and S23 decreased to 3.5 dB, as shown in Figure 4(f-h).

S6. Explanation of isolation ratio change in terms of scattering parameters
Consequently, the ON/OFF ratio of the interference increased again, reaching 10 dB. The correlation between the differences between S21 and S23, and the isolation ratio is illustrated in Figure S6 (i-j).

S7. Proposed reconfigurable interference device under uniform heating by Peltier device
Our sample substrate was placed on a Peltier device to raise the temperature of the interference device uniformly, as shown in the figure below. In this way, the contribution of the temperature gradient was nullified. Figure S7: Reconfigurable interference device under uniform heating by Peltier device. Figure S8. Calculated reconfigurable interference mimicking XNOR to XOR logic operation, considering a uniform temperature difference of 3 K.